Remark 3  Observe that our analysis also work for the case \(q=1\)  and \(\alpha >3.\)  Indeed, in this case by taking \(2<b<\alpha -1\)  and \(0<K<\min \left(b-2,\alpha -1-b,\frac{a}{\alpha +b-1}\right)\)  in the proof of Theorem REF  we reobtain some results from [1]} and [2]}.
More precisely if \(p\le 2\)  one has
\(g\left(x(t)\right) - \min g =O\left(\frac{1}{t^{p}}\right) \mbox{ as }t\rightarrow +\infty \mbox{ and } \Vert  \dot{x}(t) \Vert =O\left(\frac{1}{t^{\frac{p}{2}}}\right) \mbox{ as }t\rightarrow +\infty .\) 
